###### Advertisements

###### Advertisements

The decay constant of `""_80^197`Hg (electron capture to `""_79^197`Au) is 1.8 × 10^{−4} S^{−1}. (a) What is the half-life? (b) What is the average-life? (c) How much time will it take to convert 25% of this isotope of mercury into gold?

###### Advertisements

#### Solution

Given :-

Decay Constant of `""_80^197"Hg" , lambda = 1.8 xx 10^-4 "s"^-1`

(a)

Half-life, `T_"1/2" = 0.693/lambda`

`⇒ T_"1/2" = 0.693/(1.8 xx 10^-4)`

= 3850 s=64 minutes

(b)

Average life, `T_(av) = T_"1/2"/0.693`

`= 64/0.693`

= 92 minutes

(c)

Number of active nuclei of mercury at t = 0 = N_{0} = 100

Active nuclei of mercury left after conversion of 25% isotope of mercury into gold = N = 75

Now , `N/N_0 = e^(-lambda t)`

Here,

N = Number of inactive nuclei

`N_0` = Number of nuclei at t = 0

`lambda =` Disintegration constant

On substituting the values, we get

`75/100 = e^(-lambdat)`

`⇒ 0.75 = e^(-lambda x)`

`⇒ "In" 0.75 = - lambda t`

`⇒ t = ("In" 0.75)/-0.00018`

= 1600 s

#### APPEARS IN

#### RELATED QUESTIONS

The decay constant of radioactive substance is 4.33 x 10^{-4} per year. Calculate its half life period.

State the law of radioactive decay.

Derive the mathematical expression for law of radioactive decay for a sample of a radioactive nucleus

Write symbolically the process expressing the β^{+} decay of `""_11^22Na`. Also write the basic nuclear process underlying this decay.

The Q value of a nuclear reaction A + b → C + d is defined by

Q = [m_{A}+ m_{b }− m_{C }− m_{d}]c^{2} where the masses refer to the respective nuclei. Determine from the given data the Q-value of the following reactions and state whether the reactions are exothermic or endothermic.

\[\ce{^12_6C + ^12_6C ->^20_10Ne + ^4_2He}\]

Atomic masses are given to be

`"m"(""_1^2"H")` = 2.014102 u

`"m"(""_1^3"H")` = 3.016049 u

`"m"(""_6^12C)` = 12.000000 u

`"m"(""_10^20"Ne")` = 19.992439 u

Represent Radioactive Decay curve using relation `N = N_o e^(-lambdat)` graphically

A radioactive nucleus 'A' undergoes a series of decays as given below:

The mass number and atomic number of A_{2} are 176 and 71 respectively. Determine the mass and atomic numbers of A_{4} and A.

Using the equation `N = N_0e^(-lambdat)` obtain the relation between half-life (T) and decay constant (`lambda`) of a radioactive substance.

In a given sample, two radioisotopes, A and B, are initially present in the ration of 1 : 4. The half lives of A and B are respectively 100 years and 50 years. Find the time after which the amounts of A and B become equal.

A radioactive nucleus ‘A’ undergoes a series of decays according to the following scheme:

The mass number and atomic number of A are 180 and 72 respectively. What are these numbers for A_{4}?

The radioactive isotope D decays according to the sequence

If the mass number and atomic number of D_{2} are 176 and 71 respectively, what is (i) the mass number (ii) atomic number of D?

In a radioactive decay, neither the atomic number nor the mass number changes. Which of the following particles is emitted in the decay?

A freshly prepared radioactive source of half-life 2 h emits radiation of intensity which is 64 times the permissible safe level. The minimum time after which it would be possible to work safely with this source is

The decay constant of a radioactive sample is λ. The half-life and the average-life of the sample are respectively

The masses of ^{11}C and ^{11}B are respectively 11.0114 u and 11.0093 u. Find the maximum energy a positron can have in the β*-decay of ^{11}C to ^{11}B.

(Use Mass of proton m_{p} = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_{n} = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c^{2},1 u = 931 MeV/c^{2}.)

The decay constant of ^{238}U is 4.9 × 10^{−18} S^{−1}. (a) What is the average-life of ^{238}U? (b) What is the half-life of ^{238}U? (c) By what factor does the activity of a ^{238}U sample decrease in 9 × 10^{9} years?

^{57}Co decays to ^{57}Fe by β^{+}- emission. The resulting ^{57}Fe is in its excited state and comes to the ground state by emitting γ-rays. The half-life of β^{+}- decay is 270 days and that of the γ-emissions is 10^{−8} s. A sample of ^{57}Co gives 5.0 × 10^{9} gamma rays per second. How much time will elapse before the emission rate of gamma rays drops to 2.5 × 10^{9}per second?

A radioactive isotope is being produced at a constant rate dN/dt = R in an experiment. The isotope has a half-life t_{1}_{/2}. Show that after a time t >> t_{1}_{/2} the number of active nuclei will become constant. Find the value of this constant.

Consider the situation of the previous problem. Suppose the production of the radioactive isotope starts at t = 0. Find the number of active nuclei at time t.

Obtain a relation between the half-life of a radioactive substance and decay constant (λ).

What is the amount of \[\ce{_27^60Co}\] necessary to provide a radioactive source of strength 10.0 mCi, its half-life being 5.3 years?

Disintegration rate of a sample is 10^{10} per hour at 20 hours from the start. It reduces to 6.3 x 10^{9} per hour after 30 hours. Calculate its half-life and the initial number of radioactive atoms in the sample.

The isotope \[\ce{^57Co}\] decays by electron capture to \[\ce{^57Fe}\] with a half-life of 272 d. The \[\ce{^57Fe}\] nucleus is produced in an excited state, and it almost instantaneously emits gamma rays.

(a) Find the mean lifetime and decay constant for ^{57}Co.

(b) If the activity of a radiation source ^{57}Co is 2.0 µCi now, how many ^{57}Co nuclei does the source contain?

c) What will be the activity after one year?

A source contains two species of phosphorous nuclei, \[\ce{_15^32P}\] (T_{1/2} = 14.3 d) and \[\ce{_15^33P}\] (T_{1/2} = 25.3 d). At time t = 0, 90% of the decays are from \[\ce{_15^32P}\]. How much time has to elapse for only 15% of the decays to be from \[\ce{_15^32P}\]?

Before the year 1900 the activity per unit mass of atmospheric carbon due to the presence of ^{14}C averaged about 0.255 Bq per gram of carbon.

(a) What fraction of carbon atoms were ^{14}C?

(b) An archaeological specimen containing 500 mg of carbon, shows 174 decays in one hour. What is the age of the specimen, assuming that its activity per unit mass of carbon when the specimen died was equal to the average value of the air? The half-life of ^{14}C is 5730 years.

Obtain an expression for the decay law of radioactivity. Hence show that the activity A(t) =λN_{O} e^{-λt}.

Two radioactive materials X_{1} and X_{2} have decay constants 10λ and λ respectively. If initially, they have the same number of nuclei, then the ratio of the number of nuclei of X_{1} to that of X_{2} will belie after a time.

A radioactive element disintegrates for an interval of time equal to its mean lifetime. The fraction that has disintegrated is ______

Which one of the following nuclei has shorter meant life?

'Half-life' of a radioactive substance accounts for ______.

The half-life of a radioactive sample undergoing `alpha` - decay is 1.4 x 10^{17} s. If the number of nuclei in the sample is 2.0 x 10^{21}, the activity of the sample is nearly ____________.

After 1 hour, `(1/8)^"th"` of the initial mass of a certain radioactive isotope remains undecayed. The half-life of the isotopes is ______.

Two radioactive materials Y_{1} and Y_{2} have decay constants '5`lambda`' and `lambda` respectively. Initially they have same number of nuclei. After time 't', the ratio of number of nuclei of Y_{1} to that of Y_{2 }is `1/"e"`, then 't' is equal to ______.

What percentage of radioactive substance is left after five half-lives?

Two electrons are ejected in opposite directions from radioactive atoms in a sample of radioactive material. Let c denote the speed of light. Each electron has a speed of 0.67 c as measured by an observer in the laboratory. Their relative velocity is given by ______.

The half-life of a radioactive nuclide is 20 hrs. The fraction of the original activity that will remain after 40 hrs is ______.

If 10% of a radioactive material decay in 5 days, then the amount of original material left after 20 days is approximately :

The half-life of the radioactive substance is 40 days. The substance will disintegrate completely in

Which sample, A or B shown in figure has shorter mean-life?

Sometimes a radioactive nucleus decays into a nucleus which itself is radioactive. An example is :

\[\ce{^38Sulphur ->[half-life][= 2.48h] ^{38}Cl ->[half-life][= 0.62h] ^38Air (stable)}\]

Assume that we start with 1000 ^{38}S nuclei at time t = 0. The number of ^{38}Cl is of count zero at t = 0 and will again be zero at t = ∞ . At what value of t, would the number of counts be a maximum?

The activity R of an unknown radioactive nuclide is measured at hourly intervals. The results found are tabulated as follows:

t (h) | 0 | 1 | 2 | 3 | 4 |

R (MB_{q}) |
100 | 35.36 | 12.51 | 4.42 | 1.56 |

- Plot the graph of R versus t and calculate the half-life from the graph.
- Plot the graph of ln `(R/R_0)` versus t and obtain the value of half-life from the graph.

The radioactivity of an old sample of whisky due to tritium (half-life 12.5 years) was found to be only about 4% of that measured in a recently purchased bottle marked 10 years old. The age of a sample is ______ years.

What is the half-life period of a radioactive material if its activity drops to 1/16^{th} of its initial value of 30 years?

The half-life of `""_82^210Pb` is 22.3 y. How long will it take for its activity 0 30% of the initial activity?